mackey functors, induction from restriction functors and coinduction from transfer functors

Journal of Algebra 315 (2007) 224–248

www.elsevier.com/locate/jalgebra

Mackey functors, induction from restriction functorsand coinduction from transfer functors

Olcay Coskun 1

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

Received 17 June 2006

Available online 18 May 2007

Communicated by Michel Broué

Abstract

Boltje’s plus constructions extend two well-known constructions on Mackey functors, the fixed-pointfunctor and the fixed-quotient functor. In this paper, we show that the plus constructions are induction andcoinduction functors of general module theory. As an application, we construct simple Mackey functorsfrom simple restriction functors and simple transfer functors. We also give new proofs for the classificationtheorem for simple Mackey functors and semisimplicity theorem of Mackey functors.© 2007 Elsevier Inc. All rights reserved.

Keywords: Mackey functors; Restriction functors; Transfer functors; Plus constructions; Fixed-point functor; Induction;Coinduction; Simple Mackey functors

1. Introduction

The theory of Mackey functors was introduced by Green to provide a unified treatmentof group representation theoretic constructions involving restriction, conjugation and transfer.Thévenaz and Webb improved Green’s definition of a Mackey functor, and they realized Mackeyfunctors as representations of the Mackey algebra μR(G). Using this identification, Thévenazand Webb applied methods of module theory to classify the simple Mackey functors [11] andto describe the structure of Mackey functors [12]. Their description of simple Mackey functors

E-mail address: [email protected] The author is partially supported by TÜBÍTAK through PhD award program.

0021-8693/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2007.04.026

O. Coskun / Journal of Algebra 315 (2007) 224–248 225

used induction and inflation from subgroups and two dual constructions, known as the fixed-pointfunctor and the fixed-quotient functor.

Applying the notion of Mackey functors to the problem of finding an explicit version ofBrauer’s induction theorem, Boltje introduced the theory of canonical induction [6,7]. In orderto solve the problem in this general context, Boltje considered not only the category MackR(G)

of Mackey functors, but also two more categories, namely the category ConR(G) of conjugationfunctors and the category ResR(G) of restriction functors. His main tools were the lower-plusand the upper-plus constructions, which extend the fixed-quotient and the fixed-point functors,respectively.

The lower-plus construction, denoted by −+, is defined as a functor ResR(G) → MackR(G).By introducing the restriction algebra ρR(G), written ρ when R and G are understood, werealize the restriction functors as representations of the restriction algebra. This leads us to

Theorem 5.1. The functors −+ and indμρ are naturally equivalent.

On the other hand, the upper-plus construction, denoted by −+, is defined as a functorConR(G) → MackR(G). By introducing the transfer algebra τR(G), written τ , and its repre-sentations, called transfer functors and realizing conjugation functors as representations of theconjugation algebra γR(G), written γ , we prove

Theorem 5.2. The functors −+ and coindμτ infτγ are naturally equivalent.

As a consequence of these identifications, we realize the fixed-point and fixed-quotient func-tors as coinduced and induced modules, respectively. Given an RG-module V , we denote by FQV

the fixed-quotient functor and by FPV the fixed-point functor.

Proposition 5.4. Let V be an RG-module. Then, the following isomorphisms hold.

(i) FQV � indμρ infργ DV and (ii) FPV � coindμ

τ infτγ DV

where DV denotes the γ -module which is non-zero only at the trivial group and DV (1) = V .

We also prove that the Brauer quotient (also known as the bar construction) is the compositionof certain restriction and deflation functors (see Corollary 5.7). Via this identification, we seethat Thévenaz’ twin functor is the composition of coinduction, inflation, deflation and restrictionfunctors.

The plus constructions are also used by Bouc [4] and Symonds [9]. To obtain informa-tion about projective Mackey functors, Bouc considered restriction functors defined only onp-subgroups and also the functor −+ (which is denoted by I in [4]). In [9], Symonds con-structed induction formulae using the plus constructions described in terms of the zero degreegroup homology and group cohomology functors.

The subalgebra structure of the Mackey algebra, we describe above, leads us to

Theorem 3.2 (Mackey structure theorem). The τ–ρ-bimodule τμρ is isomorphic to τ ⊗γ ρ.

As a consequence of this theorem, we obtain several equivalences relating the functors be-tween the algebras μ,τ,ρ and γ . Using some of these equivalences, we show that the well-

226 O. Coskun / Journal of Algebra 315 (2007) 224–248

known mark homomorphism corresponds to the identity map on conjugation functors (seeProposition 5.10).

Our module theoretic approach not only unearths the nature of some known constructions forMackey functors, but also allows us to understand the classification of simple Mackey functorsbetter. The classification theorem of Thévenaz and Webb [11] asserts that the simple Mackeyfunctors are parameterized by the G-classes of simple pairs (H,V ) where H is a subgroup of G

and V is a simple RNG(H)/H -module. It is easy to see that the simple conjugation functors arealso parameterized by the G-classes of simple pairs (H,V ). It is almost as easy that the simplerestriction functors and the simple transfer functors are parameterized in the same way. As anapplication of our characterization of the plus constructions, we show how the classification the-orem for simple Mackey functors follows quickly from the classification of the simple restrictionfunctors. Moreover, we obtain two new descriptions of the simple Mackey functors. In the casewhere |G| is invertible in the base field R, we see that induction from the restriction algebra andcoinduction from the transfer algebra respect simple modules. We also give a new proof of thesemisimplicity theorem [11], which states that the Mackey functors are semisimple when R is afield of characteristic coprime to |G|.

Let us mention that, in a sequel to this paper, we shall be adapting some of these methods andresults to the content of biset functors.

The organization of the paper is as follows. In Section 2, we collect together necessary factsconcerning the Mackey functors. In Section 3 we prove the Mackey structure theorem and itsconsequences. Section 4 contains the duality theorems. Our main results, the description of plusconstructions via induction, coinduction and restriction are proved in Section 5. Also in thissection, we give alternative descriptions of the fixed-point functor, the fixed-quotient functor,the twin functor and the mark homomorphism. The applications to the classification of simpleMackey functors and to the semisimplicity of Mackey functors are the contents of Sections 6and 7, respectively.

2. Preliminaries

Let G be a finite group and R be a commutative ring with unity. Consider the free algebra onvariables cH

g , rHK , tHK where K � H � G and g ∈ G. We define the Mackey algebra μR(G) for

G over R as the quotient of this algebra by the ideal generated by the following six relations,where L � K � H � G and h ∈ H and g,g′ ∈ G:

(1) cHh = rH

H = tHH ,

(2) cgHg′ cH

g = cHg′g and rK

L rHK = rH

L and tHK tKL = tHL ,

(3) cKg rH

K = rgHgK cH

g and cHg tHK = t

gHgK cK

g ,

(4) rHJ tHK = ∑

x∈J\H/K tJJ∩xKcxrKJx∩K for J � H (Mackey relation),

(5)∑

H�G rHH = 1,

(6) all other products of generators are zero.

It is known that, letting H and K run over the subgroups of G and letting g run over thedouble coset representatives HgK ⊂ G and letting L run over representatives of the subgroupsof Hg ∩ K up to conjugacy, the elements tHgL cL

g rKL run (without repetitions) over the elements

of an R-basis for the Mackey algebra μR(G) (cf. [12, Section 3]).


We denote by ρR(G), called the restriction algebra for G over R, the subalgebra of theMackey algebra generated by cH

g and rHK for K � H � G and g ∈ G. We denote by τR(G)

the transfer algebra for G over R the subalgebra generated by cHg and tHK for K � H � G

and g ∈ G. The conjugation algebra, denoted γR(G), is the subalgebra generated by the ele-ments cH

g . When there is no ambiguity, we write μ = μR(G), and ρ = ρR(G) and τ = τR(G)

and γ = γR(G). Evidently, the restriction algebra ρ has generators cJg rK

J , the transfer algebra τ

has generators cKg tKJ and the conjugation algebra γ has generators cJ

g .We define a Mackey functor for G over R to be a μR(G)-module. Similarly, we define a

restriction functor, a transfer functor and a conjugation functor as a ρR(G)-module, a τR(G)-module and a γR(G)-module, respectively.

We can also define a Mackey functor as a quadruple (M,c, r, t) consisting of a family ofR-modules M(K) for each K � G and families of three types of maps:

(i) conjugation maps, cKg :M(K) → M(gK) for each g ∈ G and K � G,

(ii) restriction maps, rKL :M(K) → M(L) for each L � K � G, and

(iii) transfer maps, tKL :M(L) → M(K) for each L � K � G.

These maps have to satisfy the relations (2), (3) and (4), above and the following relation

(1′) cHh = rH

H = tHH = idH for all h ∈ H � G.

We write M for the quadruple (M,c, r, t). Then to pass from the first definition to the secondone, we put M(K) = cK

1 M for each K � G and conversely, we take M = ⊕K�G M(K). Similar

comments apply to restriction and transfer and conjugation functors (cf. [8,12]).Defining a morphism of Mackey functors to be an R-module homomorphism compatible with

conjugation, restriction and transfer maps, we obtain the category MackR(G) of Mackey functorsfor G over R. Similarly, we have the category ResR(G) of restriction functors, the categoryTranR(G) of transfer functors and ConR(G) of conjugation functors.

Remark 2.1. In [4], Bouc introduced an algebra, denoted rμR(G), which is generated by cGg and

rHK where K � H � G, g ∈ G and H is a p-subgroup. He also introduced tμR(G) as the dual

of rμR(G). Upper and lower plus constructions are also introduced in this settings.

In [7,8], Boltje introduced two functors −+ : ConR(G) → MackR(G) and −+ : ResR(G) →MackR(G), called upper-plus and lower-plus constructions, respectively. In Section 5, we showthat these functors have descriptions as induction and coinduction functors. We review the con-structions of these functors.

To a conjugation functor C, we associate a Mackey functor C+ where for H � G, we definethe modules as

C+(H) =( ∏

L�H

C(L)

)H

.

Here H acts on the product by coordinate-wise conjugation. We define the maps for K � H � G

and g ∈ G and xL ∈ C(L) as follows:


Conjugation:

c+Hg :C+(H) → C+(

gH)

where (xL)L�H → (gxLg

)L�gH

.

Restriction:

r+HK :C+(H) → C+(K) where (xL)L�H → (xL)L�K.

Transfer:

t+HK :C+(K) → C+(H) where (xL)L�K →

∑h∈H/K

c+Kh

((xL)L�K

).

The functor −+ is defined on morphisms, in the obvious way, that is, if f : B → C is a morphismof conjugation functors, then f + :B+ → C+ is defined by f +

H ((xL)L�H ) = (fL(xL))L�H .To a restriction functor D, we associate a Mackey functor D+ where for H � G, the modules

are

D+(H) =( ⊕

L�H

D(L)

)H

.

Here, for an RH -module M , we write MH for the (maximal) H -fixed quotient, that is to say,MH = M/I (RH)M where I (RH) denotes the augmentation ideal of RH . For K � H and a ∈D(K), we write the image of a in D+(H) as [K,a]H . Clearly, [K,a]H = [hK, ha]H for h ∈ H

and as an R-module, D+(H) is generated by the elements [K,a]H for K � H and a ∈ D(K).The maps are defined for L � H � G and g ∈ G as follows:

Conjugation:

cH+g :D+(H) → D+(gH

)where [K,a]H → [

gK, ga]gH

.

Restriction:

rH+L :D+(H) → D+(L) where [K,a]H →∑

h∈L\H/K

[L ∩ hK, r

hKL∩hK

(ha)]

L.

Transfer:

tH+L :D+(L) → D+(H) where [N,b]L → [N,b]H .

For a morphism f :D → E of restriction functors, we define f+ :D+ → E+ byf+H ([K,a]H ) = [K,fK(a)]H for K � H and a ∈ D(K).

The plus constructions are related to each other by a morphism, called the mark homomor-phism, denoted by ρ in [7,8]. We write β for the mark homomorphism. It is defined as follows:Let D be a restriction functor and H � G. Then

βH := (πK ◦ rH+K

):D+(H) → (FD)+(H)

K�H


where F : ResR(G) → ConR(G) is the forgetful functor and πK is the projection

πK [L,a]K = a

if L = K and equal to zero otherwise. The mark homomorphism is an isomorphism if |G| isinvertible in R and is injective if D+(H) has trivial |H |-torsion for all H � G (cf. [7, Proposi-tion 1.3.2]).

The functors −+ and −+ have crucial use in constructing canonical induction formulae forMackey functors. For further details, see [7,8], for applications see [4,9].

Two other constructions in the theory of Mackey functors that are used frequently are the barconstruction and the twin functor. We review the definitions of these constructions.

Definition 2.2. Let M be a Mackey functor. The bar construction of M is the conjugation functorM where for K � G, we have

M(K) = M(K)/ ∑

L<K

Im(tKL

)

and the conjugation maps are inherited from those of M .

The bar construction composed with the functor −+ gives the twin functor T M of M (cf. [7,Section 1.1.2]). We have the following morphism between a Mackey functor and its twin. ForK � G and m ∈ M(K), we define

βK :M(K) → T M(K)

where βK(m) = (πL(rKL m))L�K and

πK :M(K) → M(K)

is the quotient map. Note that the mark homomorphism is a special case of the morphismβ :M → T M where we put M = D+ for a restriction functor D.

Let E and G be rings and α :E → G be a unital ring homomorphism. We can regard anyG-module as an E-module by α. This induces a functor

resα :G-mod → E-mod

called the generalized restriction. There are two functors in the opposite direction.Induction: We regard G as a right E-module by f e = f α(e) for e ∈ E and f ∈ G. Then, for

any (left) E-module M , we make G⊗E M a (left) G-module by f (f ′ ⊗m) = ff ′ ⊗m for m ∈ M .Note that, the action is well-defined as the natural action of G on itself commutes with the actionof E on G. We call G ⊗E M the induced module, written indα M , and obtain the generalizedinduction functor

indα − := G ⊗E − :E-mod → G-mod.

Coinduction: Now we regard G as a left E-module by ef = α(e)f for e ∈ E and f ∈ G. Then,for any (left) E-module M , we make HomE (G,M) a (left) G-module by (f φ)(f ′) = φ(f ′f ) for


f,f ′ ∈ G and φ ∈ HomE (G,M). Note that, the natural action of G on itself commutes with theaction of E on G. We call HomE (G,M) the coinduced module, written coindα M , and obtain thegeneralized coinduction functor

coindα := HomE (G,−) :E-mod → G-mod.

We recall the adjointness properties of these three functors:

Proposition 2.3. The induction functor indα is right adjoint of the restriction resα . The coinduc-tion functor coindα is the left adjoint of the restriction resα .

The proof of the proposition and further details can be found in [2, Section 3.3]. In all ourapplications, α will be an inclusion E ↪→ G or a projection E → G = E/Δ for some ideal Δ of E .For the first case, we write the induction and coinduction functors as indGE and coindGE , respec-tively. For the second case, we write induction and restriction as defEG and infEG , respectively.

Finally, we recall the following well-known proposition.

Proposition 2.4. (See [1, Section 2.8].) Let E and G be rings. Let M be a left G-module and letA be a G–E-bimodule and let N be a left E-module. Then, there is a natural isomorphism

HomE(N,HomG(A,M)

) ∼= HomG(A ⊗E N,M).

3. The Mackey triangle

In this section, we examine the relations between the algebras μ, τ , ρ and γ . Mainly, weexplain the following triangle, which we call the Mackey triangle.

μ

τ ρ

γ γ γ

Here the arrows ρ ↪→ μ and τ ↪→ μ denote the inclusions of algebras and so are γ → ρ andγ → τ . The arrows ρ � γ and τ � γ denote surjections explained in the next lemma, whichalso describes the identifications at the bottom of the triangle.

Lemma 3.1. Let J (ρ) be the two-sided ideal of the restriction algebra ρ generated by all non-trivial restriction maps. Then, there is an evident identification γ = ρ/J (ρ). Similarly, we makethe identification γ = τ/J (τ ) where J (τ ) is generated by all non-trivial transfer maps.

Proof. Recall that the restriction algebra (respectively transfer algebra) is generated by cgrKH

where H � K � G and g ∈ G. As an R-module, J (ρ) is spanned by the elements cgrKH where

K < H . It is now clear that the quotient is isomorphic to the conjugation algebra. The last partcan be proved similarly. �


The main property of the Mackey triangle is the following.

Theorem 3.2 (Mackey structure theorem). The τ–ρ-bimodule τμρ is isomorphic to τ ⊗γ ρ.

Proof. It is clear that τ ⊗γ ρ is generated by the elements tHgJ ⊗cJg rK

J . Now we show that τ ⊗γ ρ

is freely generated by these elements. To this aim, we decompose the left γ -module ρ as

γ ρ =⊕

K�G,J�KK

γ rKJ

and the right γ -module τ as

τγ =⊕

H�G,I�H H

tHI γ.

Then, the tensor product becomes

τ ⊗γ ρ =⊕

I�H H�G,J�KK�G

tHI γ ⊗γ γ rKJ

=⊕

I�H H�G,J�KK�G,L�GG

RtHI γ c[L] ⊗γ c[L]γ rKJ

where c[L] = ∑L′=GL cL′

. Here cL is the generator cL1 .

To focus on each summand separately, fix H,K,L � G. Then,

c[L]γ rKJ =

⊕x∈G/K,Lx=KJ

RcLcJx rK

J .

Indeed, the equality holds since J is taken up to K-conjugacy and cLx cJ = 0 unless Lx =K J .

Similarly,

tHI γ c[L] =⊕

y∈H\G,yL=H I

RtHI cLy cL.

Hence,

tHI γ c[L] ⊗γ c[L]γ rKJ =

⊕x,y

RtHI cLy ⊗ cJ

x rKJ .

Therefore,

τ ⊗γ ρ =⊕

H,K,I,J,L,x,y

RtHI cLy ⊗ cJ

x rKJ

=⊕ ⊕

g

RtHI ⊗ cJg rK

J .

H,K,I,J g∈H\G/K,I =J


Hence, we see that τ ⊗γ ρ is freely generated over R by the elements tHgJ ⊗ cJg rK

J . It is also clear

from the last equation that given tHgJ ⊗ cJg rK

J and tHf I⊗ cI

f rKI then

tHgJ ⊗ cJg rK

J = tHf I⊗ cI

f rKI

if and only if HgK = Hf K and J and I are Hg ∩ K-conjugate. But this is equivalent to sayingthat tHgJ cJ

g rKJ is equal to tHf I

cIf rK

I as elements of the Mackey algebra (cf. [12, Proposition 3.2]).Hence the correspondence

Γ : τ ⊗γ ρ → μ

given by Γ (tHgJ ⊗cJg rK

J ) = tHgJ cJg rK

J extends linearly to an isomorphism of R-modules. Evidently,the map Γ is compatible with the left action of the transfer algebra τ and the right action of therestriction algebra ρ. Thus Γ is an isomorphism of τ–ρ-bimodules from τ ⊗γ ρ to τμρ . �

Now as a result of these relations we obtain several induction, coinduction and restrictionfunctors and some equivalences between them. As we shall see in the next section, some ofthese functors are also naturally equivalent to some well-known constructions. For the rest ofthis section, we prove some equivalences as consequences of Theorem 3.2. In the next lemma,which we state without proof, we collect some trivial but necessary observations about some ofthese functors:

Lemma 3.3. In the Mackey triangle, there are two inflation functors, infτγ and infργ . For a γ -module C, the τ -module infτγ C is the module C regarded as a τ -module by letting all non-trivial

transfer maps tKL for L < K � G act as zero maps. A similar result holds for the ρ-moduleinfργ C. Moreover, the compositions resτ

γ infτγ and resργ infργ are both naturally equivalent to the

identity functor on γ -mod.

For the rest of this section, we prove more equivalences. Most of the equivalences are conse-quences of the Mackey structure theorem.

Theorem 3.4. The following natural equivalences hold.

(i) indτγ resρ

γ∼= resμ

τ indμρ .

(ii) coindργ resτ

γ∼= resμ

ρ coindμτ .

Proof. The first equivalence is induced by the isomorphism Γ of τ −ρ-bimodules μ and τ ⊗γ ρ

defined in the proof of Theorem 3.2. Indeed

indτγ resρ

γ∼= τ ⊗γ ρ ⊗ρ and resμ

τ indμρ

∼= τμρ ⊗ρ .

The induced equivalence is clearly natural. To prove the second equivalence, note that by thedefinition of coinduction,

coindργ resτ

γ = Homγ

(ρ,Homτ (τ,−)

).


Now applying Proposition 2.4, we obtain a natural equivalence

Υ : Homγ

(ρ,Homτ (τ,−)

) ∼= Homτ (τ ⊗γ ρ,−)

of functors with values in R-mod. It is easy to check that for any τ -module E, the isomorphismΥE is compatible with conjugation and restriction maps. But, in that case the right-hand side ofthe last equation becomes

Homτ (τ ⊗γ ρ,−) ∼= resμρ coindμ

τ

since τ ⊗γ ρ ∼= μ as left τ -modules. �Corollary 3.5. The following equivalences hold.

(i) indτγ

∼= resμτ indμ

ρ infργ .(ii) coindρ

γ∼= resμ

ρ coindμτ infτγ .

Proof. This follows from Theorem 3.4 and Lemma 3.3 by composing with the correspondinginflations. �

Finally, we have two more functors that are naturally equivalent to the identity functor onγ -mod. Let us write codefργ for the left adjoint of the inflation infργ . Explicitly, for a ρ-module D

and for K � G, we have

codefργ D(K) =⋂

L<K

Ker(rKL :D(K) → D(L)

)

and the conjugation maps are obtained from those for the ρ-module D. The other functor is thedeflation functor defτγ induced by the map of Lemma 3.1. Note also that we have a deflationfunctor defργ and a codeflation functor codefτγ , but we shall not introduce these as we will not usethem.

Proposition 3.6. The following equivalences hold:

defτγ indτγ

∼= idγ∼= codefργ coindρ

γ .

Proof. The equivalences follows easily from Lemma 3.3, since a left and a right adjoint of theidentity functor and the identity functor are naturally equivalent to each other. �4. Duality theorems

Theorem 3.4 suggests a duality in the Mackey triangle. In this section, we clarify this duality.Following [11], we denote by −op, the opposite functor, defined by

−op :μ-mod → mod-μ


where for a left μ-module M , the right μ-module Mop is the same R-module M with the rightMackey functor structure given by

m(tHgJ cgr

KJ

) = (tKJ cg−1r

HgJ

)m

where tHgJ cgrKJ ∈ μ and m ∈ M(H).

We have another duality (cf. [11])

Dμ :μ-mod → mod-μ

where for a left μ-module M , we let DμM to be the right μ-module HomR(M,R) where μ actson the right as usual. Note that DμM is the usual duality D∗ in module theory. Clearly, thesefunctors can be defined in the reverse direction, and we can compose one with the other to obtain

Dopμ :μ-mod → μ-mod.

Note that there is no ambiguity writing Dopμ since the functors commute.

The functors DμM and −op also induce functors on the modules of the subalgebras ρ and τ .Since −op interchanges restriction and transfer maps, we obtain dualities

−op :ρ-mod → mod-τ and −op : τ -mod → mod-ρ.

On the other hand, the functor Dμ induces

Dρ :ρ-mod → mod-ρ and Dτ : τ -mod → mod-τ.

The following theorem describes induction from right τ -modules and coinduction from rightρ-modules to right μ-modules.

Theorem 4.1 (The first duality theorem). Let D be a ρ-module and E be a τ -module. Then

(i) (indμρ D)op ∼= indμ

τ (Dop) where indμτ : mod-τ → mod-μ.

(ii) (coindμτ E)op ∼= coindμ

ρ (Eop) where coindμρ : mod-ρ → mod-μ.

Proof. The first part is clear, since we have

(indμ

ρ D)op = (μ ⊗ρ D)op ∼= Dop ⊗τ μ = indμ

τ

(Dop).

The second part can be proved similarly. �Combining the above functors, we can define

Definition 4.2. The transfer-restriction duality is the equivalence

Dopρ := D−1

ρ ◦ −op : τ -mod → ρ-mod

of categories τ -mod ∼= ρ-mod. We call the inverse equivalence

Dopτ := D−1

τ ◦ −op :ρ-mod → τ -mod

the restriction-transfer duality.


Finally, note that Dopμ induces a duality Dop

γ on γ -modules. The following theorem describesthe duality we promised earlier.

Theorem 4.3 (Restriction-transfer duality). Let D be a ρ-module and E be a τ -module. Then

(i) Dopμ (indμ

ρ D) ∼= coindμτ (Dop

τ D).

(ii) Dopμ (coindμ

τ E) ∼= indμρ (Dop

ρ E).

Proof. The first part follows from Proposition 2.3 as we have

Dopμ

(indμ

ρ D) = HomR

((indμ

ρ D)op

,R)

= HomR

(indμ

τ

(Dop),R)

∼= Homτ

(μ,HomR

(Dop,R

))= coindμ

τ

(Dop

τ D).

Note that although the above isomorphism is an isomorphism of R-modules, it is easily checkedthat it is an isomorphism of left Mackey functors. The second statement can be proved simi-larly. �

In the next theorem, we collect together some more dualities relating induction, coinductionand restriction. The theorem and any other duality can be proved in the same way.

Theorem 4.4. Let M be a μ-module, E be a τ -module and C be a γ -module. Then

(i) Dopτ (resμ

ρ M) ∼= resμτ (Dop

μ M).

(ii) Dopρ (resμ

τ M) ∼= resμρ (Dop

μ M).

(iii) Dopτ (infργ C) ∼= infτγ (Dop

γ C).

(iv) Dopρ (infτγ C) ∼= infργ (Dop

γ C).

(v) Dopγ (defτγ E) ∼= codefργ (Dop

ρ E).

Let us end with an abstraction of the above situation. Let Υ be a finitely generated R-algebrasuch that it has subalgebras Υ↑, Υ↓ and Υ− with the following two properties.

(i) The subalgebras together with Υ form the following triangle:

Υ

Υ↑ Υ↓

Υ− Υ− Υ−

where the maps are as explained in the previous section.(ii) The structure theorem, Υ↑ΥΥ↓ � Υ↑ ⊗Υ− Υ↓, holds.


Then the results in Sections 3 and 4 hold for the modules of the algebras Υ , Υ↑, Υ↓ and Υ− andfor induction, coinduction and restriction functors. Moreover our classification and descriptionof simple Mackey functors can be modified for the simple modules of the algebra Υ .

There are at least two more algebras having this structure. The first example is the algebraμA associated to a Green functor A (see [3] for the definition). Note that the Mackey algebra isobtained by taking A = BG, the Burnside Mackey functor [3].

Another occurrence of this structure is in the biset functors, introduced by Bouc [5]. As men-tioned in the introduction we shall adopt the methods of this paper to the analogous algebra forbiset functors.

5. Plus constructions via induction and coinduction

In this section, we show that under the equivalence of categories μR(G) ∼= MackR(G), theplus constructions −+ and −+ are realizable in terms of generalized restriction, generalizedinduction and generalized coinduction. Moreover the well-known fixed-point functor and thefixed-quotient functor [11], and the twin functor [10] have similar descriptions. We begin byproving our first identification.

Theorem 5.1. The functors −+ and indμρ are naturally equivalent.

Proof. To specify a natural equivalence Φ : indμρ → −+, we must specify a map of μ-modules

φD : indμρ D → D+

for any ρ-module D and show that it is natural in D. To do that, we must specify an isomorphismof R-modules

φD,H : indμρ D(H) → D+(H)

for any subgroup H � G which is compatible with the actions of transfer, restriction and conju-gation. Now

indμρ D(H) =

⊕K�H H

{tHK ⊗ρ a: a ∈ D(K)

}

where the notation indicates that K runs over representatives of the conjugacy classes of sub-groups of H . Also,

D+(H) =⊕

K�H H

{[K,a]H : a ∈ D(K)}.

We have tHK ⊗ aK = 0 if and only if aK ∈ I (NH (K))D(K), where I (NG(H)) is the augmenta-tion ideal as before. But this is equivalent to the condition that [K,a]H = 0. So we can defineΦD,H by

ΦD,H

(tHK ⊗ρ a

) = [K,a]H .


Thus, we have defined an R-module isomorphism ΦD = (ΦD,H )H�G from indμρ D to D+. Now

we show that ΦD is compatible with the actions of conjugation, restriction and transfer. We mustalso check that Φ is natural.

Given L � G and a ∈ D(K), then

ΦD,L

(rHL

(tHK ⊗ a

)) = ΦD,L

( ∑h∈L\H/K

tLL∩hK

rhKL∩hK

cKh ⊗ a

)

=∑

h∈L\H/K

ΦD,L

(tLL∩hK

⊗ rhKL∩hK

cKh a

)

=∑

h∈L\H/K

[L ∩ hK, r

hKL∩hK

cKh a

]L

= rH+L[K,a]H= rH+LΦD,H

(tHK ⊗ a

).

We have established compatibility with restriction, ΦDrHL = RH

L ΦD . Compatibility with conju-gation and transfer can be shown similarly (and more easily).

Finally, for the naturality, consider a map of ρ-modules f : D → D′. The maps of μ-modules

indμρ f : indμ

ρ D → indμρ D′, f+ :D+ → D′+

are given by (indμρ f )H (tHK ⊗ a) = tHK ⊗ fK(a) and (f+)H ([K,a]H ) = [K,fK(a)]H . Hence,

ΦD′(indμ

ρ f(tHK ⊗ a

)) = φD′(tHK ⊗ fK(a)

) = [K,fK(a)

]H

= f+([K,a]H

)= f+

(ΦD

(tHK ⊗ a

)).

So ΦD′ ◦ indμρ f = f+ ◦ ΦD , in other words, Φ is natural. �

Theorem 5.2. The functors −+ and coindμτ infτγ are naturally equivalent.

Proof. As in the previous proof, to specify a natural equivalence Ψ : coindμτ infτγ → −+, we must

specify a map of μ-modules

ΨC : coindμτ infτγ C → C+

for any γ -module C and show that it is natural in C. In order to do that, we must specify anR-module isomorphism

ΨC,H : coindμτ infτγ C(H) → C+(H)

for any subgroup H � G and show that it is compatible with the action of transfer, restrictionand conjugation. Now

coindμτ infτγ C(H) = Homτ

(μ, infτγ C

)(H) = Homτ

(μcH , infτγ

)


where infτγ C = ⊕J�H C(J ). Recall that any element of μcH is a linear combination of ele-

ments of the form tKgJ cgrHJ where g ∈ G and J � Kg ∩ H . But, for such an element and for a

map φ :μcH → infτγ C of τ -modules, we have

φ(tKgJ cgr

HJ

) = tKgJ φ(cgr

HJ

) = 0

unless K = gJ . Indeed, tKL annihilates the τ -module infτγ C if L �= K . Also, if K = gJ , then

φ(cgr

HJ

) = cgφ(rHJ

)

that is, the value of φ at cgrHJ is determined by the value of φ at rH

J . Moreover, for any h ∈ H ,we have

φ(rHhJ

) = φ(rHhJ

cHh

) = φ(cJh rH

J

) = cJh

(φ(rHJ

)).

Now recall that

C+(H) =( ∏

J�H

C(J )

)H

={(xJ )J�H ∈

∏J�H

C(J ): h(xJ ) = xhJ for J � H, h ∈ H

}.

So, we can define

ΨC,H (φ) = (φ(rHJ

))J�H

.

The map ΨC,H is an isomorphism of R-modules from coindμτ infτγ C(H) to C+(H) with the

inverse given by

Ψ −1C,H (X) = φX.

Here, X = (xJ )J�H and φX is the map defined by φX(cgrHJ ) = g(xJ ). Thus, we have defined

an R-module isomorphism ΨC : coindμτ infτγ C → C+. We must show that ΨC is compatible with

the actions of conjugation, restriction and transfer. Also, we must check that Ψ is natural in C.Given J � H � K � G and φ ∈ coindμ

τ infτγ C(H), then

ΨC,H

(tKH φ

) = ((tKH φ

)(rKJ

))J�K

= (φ(rKJ tKH

))J�K

=( ∑

x∈J\K/H

(tJJ∩xH cx

)(φ(rHJx∩H

)))J�K

=( ∑

x∈J\K/H,J∩xH=J

cx

(φ(rHJx

)))J�K

=( ∑

x∈K/H,J x�H

cx

(φ(rHJx

)))J�K

= t+K(φ(rHJ

)) = t+KΨC,H (φ).
H J�H H


We have established compatibility with transfer, ΨC,K ◦ tKH = tKH ΨC,H . Compatibility with re-striction and conjugation can be proved similarly. Finally, one can check that the transformationΨ is natural as above. �

By Theorems 3.4 and 5.2, we obtain an explicit description of the functor coindμτ .

Theorem 5.3. Let E be a transfer functor. Then for H � G, we have

coindμτ E(H) ∼=

( ∏L�H

E(L)

)H

.

The actions of conjugation and restriction are the same as the actions of conjugation and restric-tion for the functor E+, respectively, and the transfer map is defined for φ ∈ coindμ

τ E(H) andK � H as

(tKH φ

)(rKJ

) =∑

k∈J\K/H

(tJJ∩kH

ck

)(φ(rHJk∩H

)).

Proof. By Theorem 3.4, there is an isomorphism

resμρ coindμ

τ E ∼= coindργ resτ

γ E

of ρ-modules. Now by Corollary 3.5, we obtain

resμρ coindμ

τ E ∼= resμρ coindμ

τ infτγ resτγ E.

Now by Theorem 5.2, the right-hand side is (resτγ E)+ regarded as a restriction functor. Hence

the isomorphism

coindμτ E(H) ∼=

( ∏L�H

E(L)

)H

holds. Evidently, the actions of conjugation and restriction are the same as those for the right-hand side. Finally it is clear that the action of transfer is given as above. �

Given an RG-module V , we denote by DV the conjugation functor where DV (1) = V andDV (H) = 0 for 1 �= H � G.

Proposition 5.4. The following isomorphisms hold.

(i) FQV∼= indμ

ρ infργ DV .(ii) FPV

∼= coindμτ infτγ DV .

Proof. It is clear from the construction of the fixed-point functor and the fixed-quotient functorthat we have the following isomorphisms (cf. [11, Section 6]):

FPV∼= (DV )+ and FQV

∼= (infργ DV

)+.

Now the result follows from Theorem 5.1 and Theorem 5.2. �


Corollary 5.5. (See [11, Proposition 6.1].) The functor indμρ infργ DV is left adjoint to the functor

F :μ-mod → RG-mod which sends a Mackey functor M to the RG-module c11M = M(1). The

right adjoint of F is coindμτ infτγ DV .

Proof. We have infργ DV (K) = 0 for each subgroup 1 < K � G and infργ DV (1) = V . Therefore

Homμ

(indμ

ρ infργ DV ,M) ∼= Homρ

(infργ DV , resμ

ρ M)

∼= HomRG(infργ DV (1), resμ

ρ M(1))

∼= HomRG(V,M(1)

)∼= HomRG(V ,FM).

The second statement is proved similarly. �Remark 5.6. It is possible to define the fixed-point functor and the fixed-quotient functor for theright μ-modules, as well as the other constructions. For example, by the Duality Theorem 4.1,we see that

Dopμ

(indμ

ρ infργ DV

) = coindμτ infτγ

(Dop

γ DV

)

which is the part (iii) of Proposition 4.1 in [12]. Also, note that we can define a fixed-pointfunctor and a fixed-quotient functor for the right μ-modules using the functor −op. In that case,for a right RG-module V , we define

V FQ := indμτ infτγ V D.

By the Duality Theorem 4.1, we obtain

V FQ = (indμ

ρ infργ DV op)op = indμ

τ infτγ DV .

We can define V FP similarly.

Finally, we have the following proposition.

Proposition 5.7. The bar construction ?, defined in Definition 2.2 is naturally equivalent todefτγ resμ

τ .

Proof. This is immediate from the equality

(J (τ )M

)(H) =

∑L<H

tHL M(L)

for H � G. �Corollary 5.8. The twin functor T is naturally equivalent to coindμ

τ infτγ defτγ resμτ .


The morphism β between a Mackey functor and its twin can be expressed in terms of theabove equivalence.

Proposition 5.9. Let M be a Mackey functor. The morphism

β :M → coindμτ infτγ defτγ resμ

τ M

as an element in Homμ(M, coindμτ infτγ defτγ resμ

τ M) is induced by the identity endomorphismiddefτγ resμτ M in Homγ (defτγ resμ

τ M,defτγ resμτ M).

Proof. By Proposition 2.3

Homμ

(M, coindμ

τ infτγ defτγ resμτ M

) ∼= Homτ

(resμ

τ M, infτγ defτγ resμτ M

)∼= Homγ

(defτγ resμ

τ M,defτγ resμτ M

).

Now the counit of the adjunction

Homτ

(resμ


) ∼= Homγ

(defτγ resμ

τ M,defτγ resμτ M

)

is given by composition with the quotient map. That is, for φ : defτγ resμτ M → defτγ resμ

τ M , the

corresponding morphism φ : resμτ M → infτγ defτγ resμ

τ M is given by

φH (m) = φH

(πH (m)

)

where m ∈ M(H) and π : resμτ M → defτγ resμ

τ M is the quotient map.On the other hand, the counit of the adjunction

Homμ

(M, coindμ

τ infτγ defτγ resμτ M

) ∼= Homτ

(resμ


)

is given by composition with the restriction maps. Explicitly, for ψ : resμτ M → infτγ defτγ resμ

τ M ,

the corresponding morphism ψ :M → coindμτ infτγ defτγ resμ

τ M is given by

ψH (m) = (ψK

(rHK m

))K�H

where m ∈ M(H).

Now put φ = id. Then φH (m) = πH (m) is the quotient map. Then, put ψ = φ and getψH (m) = (ψK(rH

K m))K�H = (πK(rHK m))K�H , which coincides with the definition of the mor-

phism β defined in Section 2. �Since the mark homomorphism is a special case of the morphism β , we have the following

corollary.

Corollary 5.10. Let D be a ρ-module. The mark homomorphism

β : indμρ D → coindμ

τ infτγ resργ D

is induced by the identity endomorphism idresρ D of the γ -module resργ D.

γ


Proof. Let us put M = indμρ D for some ρ-module D. Then by part (i) of Theorem 3.4 and by

Proposition 3.6

coindμτ infτγ defτγ resμ

τ indμρ D ∼= coindμ

τ infτγ resργ D.

Also the quotient map π above coincides with the projection map π since defτγ resμτ M = D. �

6. Simple Mackey functors

Throughout this section, we assume that R is a field. In [11], Thévenaz and Webb establisheda bijective correspondence between the G-classes of the simple pairs (H,V ) where H � G andV a simple RNG(H)-module where NG(H) := NG(H)/H and the isomorphism classes of thesimple Mackey functors. We denote by SH,V the simple Mackey functor corresponding to thepair (H,V ), under this correspondence.

To illustrate the usefulness of our module-theoretic approach we give an alternative proof tothis result by realizing simple Mackey functors as quotients of induced simple restriction func-tors. As we shall see below, the classification of simple restriction functors is trivial. Then wegive another new description of the simple Mackey functor SH,V as the unique minimal sub-functor of the Mackey functor coindμ

τ SτH,V , where Sτ

H,V is a simple transfer functor, introducedbelow.

Throughout this section, let H � G and V be a simple RNG(H)-module. We write Sγ

H,V forthe conjugation functor defined for K � G by

Sγ

H,V (K) = gV if K = gH

and zero otherwise. We also write SρH,V = infργ Sγ

H,V and SτH,V = infτγ Sγ

H,V .

Proposition 6.1. (See [7, Remark 1.6.6], [4, Proposition 3.2].) The followings hold.

(i) The conjugation functor Sγ

H,V is simple. Moreover, any simple conjugation functor is iso-

morphic to Sγ

H,V for some simple pair (H,V ).

(ii) The restriction functor SρH,V is simple. Moreover, any simple restriction functor is isomor-

phic to SρH,V for some simple pair (H,V ).

(iii) The transfer functor SτH,V is simple. Moreover, any simple transfer functor is isomorphic to

SτH,V for some simple pair (H,V ).

We recall, without proof, the description of the simple Mackey functors SH,V := SGH,V from

[11]. Since the Mackey algebra μR(H) is a (non-unital) subalgebra of the Mackey algebraμR(G) for H � G and the Mackey algebra μR(G/H) is a quotient of the Mackey algebra μR(G)

for H � G, we obtain an induction functor indμR(G)

μR(H) and an inflation functor infμR(G)

μR(G/H). Explicitdescriptions of these functors are given in [11, Section 4,5].

Lemma 6.2. (See [11, Lemma 8.1].) Let H be a subgroup of G and V be a simple RNG(H).Then


(i) The functor M = indGNG(H) infNG(H)

NG(H)FPV has a unique minimal subfunctor SG

H,V generated

by M(H) = V .(ii) The functor indG

NG(H) infNG(H)

NG(H)FQV has a unique maximal subfunctor. Moreover, the quo-

tient is isomorphic to SGH,V .

Now we want to state the main result of this section. For this we need the following notation.Let M be a Mackey functor and H � G be a minimal subgroup for M , that is, M(L) = 0 forL < H and M(H) �= 0. After [11], we define two subfunctors of M as follows:

IM(H)(K) =∑

L�K: L=GH

Im(tKL :M(L) → M(K)

)

and

KM(H)(K) =⋂

L�K: L=GH

Ker(rKL : M(K) → M(L)

).

Theorem 6.3. We have the following isomorphisms of Mackey functors

SGH,V � indμ

ρ SρH,V /Kindμ

ρ SρH,V (H) � Icoindμ

τ SτH,V (H).

We prove the theorem in several steps. The first step is the following lemma.

Lemma 6.4. The subfunctor K = Kindμρ Sρ

H,V (H) of the Mackey functor indμρ Sρ

H,V is the unique

maximal subfunctor of indμρ Sρ

H,V .

Proof. Let T be a proper subfunctor of indμρ Sρ

H,V . We are to show that T � K, that is

T (K) ⊂⋂

L�K: L=GH

Ker(rKL

)

for any K � G. So, we must show that for each K � G and any x ∈ T (K), we haverKL x = 0 for all H =G L � K . But since indμ

ρ SρH,V (H) = V , it is evident that T (L) = 0 for

any L =G H . Indeed, otherwise T (H) = V as V is a simple RNG(H)-module. But, by defini-tion of the action of tKL , the functor indμ

ρ SρH,V is generated by the images of the transfer maps tKL

for H =G L � K , that is, we have Iindμρ Sρ

H,V (H) = indμρ Sρ

H,V . Hence the subfunctor T contain-

ing the subfunctor generated by T (H) = V is not proper, contradicting our assumption. Thus,T � K as required. �

We denote the simple quotient of indμρ Sρ

H,V by

SH,V = indμρ Sρ

H,V /K.

Note that if (K,W) is another simple pair, then SK,W is not isomorphic to SH,V . Indeed, sinceK(H) = 0, the subgroup H is still a minimal subgroup of the quotient SH,V and similarly, K is


a minimal subgroup of SK,W . Hence for K �= H , the simple modules SH,V and SK,W are non-isomorphic. Also, for K = H , any morphism SH,V → SK,W of Mackey functors induces a mapV → W of RNG(H)-modules. But, by the Schur’s lemma, any such map is either an isomor-phism or the zero map. Thus SH,V is not isomorphic to SK,W unless H = K and V ∼= W .

Having the above description, we get another proof of Thévenaz and Webb’s classificationtheorem:

Theorem 6.5. Any simple Mackey functor is isomorphic to SH,V for some simple pair (H,V ).

Proof. Let S be a simple Mackey functor with a minimal subgroup H and S(H) = V . It sufficesto show that there is a non-zero morphism of Mackey functors SH,V → S. We show that there isa morphism of Mackey functors F : indμ

ρ SρH,V → S such that FH �= 0.

By Proposition 2.4, we have

Homμ

(indμ

ρ SρH,V , S

) � Homρ

(Sρ

H,V , resμρ S

).

But, SρH,V (K) = 0 unless K =G H . So the identity map idV :V → V of RNG(H)-modules

induces a non-zero map f :SρH,V → resμ

ρ S of ρ-modules. Hence the corresponding map F ∈Homμ(indμ

ρ SρH,V , S) is non-zero. Moreover, since indμ

ρ SρH,V (H) = V , we have FH = fH =

id �= 0. Thus, the induced morphism F : SH,V → S is non-zero, as required. �Hereafter, we identify SH,V with SG

H,V and write SH,V when the group G is understood. Wecomplete the proof of Theorem 6.3 by the following lemma.

Lemma 6.6. The subfunctor I = Icoindμτ Sτ

H,V (H) generated by I(H) = V is the unique minimal

subfunctor. Moreover, the subfunctor I is isomorphic to SH,V .

Proof. Let T be a non-zero subfunctor of coindμτ Sτ

H,V . We must show that I � T . It suffices toshow that T (H) �= 0. Indeed, in that case, since coindμ

τ SτH,V (H) = V is simple, T (H) = V and

hence I � T . But Kcoindμτ Sτ

H,V (H) = 0 by the definition of the map rKH and the diagram

T (H)i

rKH

coindμτ Sτ

H,V (H)

rKH

T (K)i

coindμτ Sτ

H,V (K)

commutes for all g ∈ G. That is to say, rKH T (K) �= 0, or T (H) �= 0. The last claim follows from

the classification Theorem 6.5. �To find the modules SH,V (K) for K � G, we need to know the subfunctor K of indμ

ρ SρH,V . In

particular, when K is zero, we get a more explicit description. The following is a characterizationof the subfunctor K.


Lemma 6.7. The subfunctor K of indμρ Sρ

H,V (K) coincides with the kernel of the mark homomor-

phism β : indμρ Sρ

H,V → coindμτ Sτ

H,V . Moreover, the subfunctor I of coindμτ Sτ

H,V is the imageof β .

Proof. As K is the unique maximal subfunctor of indμρ Sρ

H,V (K), we have kerβ ⊂ K. So, itsuffices to show the inverse inclusion. Given K � G and x ∈ K(K), then

βK(x) = (ηL

(rKL x

))L�K,L=GH

= 0

since rKL x = 0 by definition of K(K). Therefore, K ⊂ kerβ . The second claim is easy since βH

is identical. �Now, using the next proposition from [7], and the above identification of the subfunctor K,

we describe K, in some cases.

Proposition 6.8. (Cf. [7, Proposition 1.3.2], [10, Section 3].) The mark homomorphism βK isinjective if indμ

ρ SρH,V (K) has trivial |K|-torsion. It is an isomorphism if |K| is invertible in R.

Corollary 6.9.

(i) If indμρ Sρ

H,V (K) has trivial |K|-torsion, then K(K) = 0.

(ii) If indμρ Sρ

H,V (K) has trivial |K|-torsion for all K � G, then indμρ Sρ

H,V is simple.

(iii) If |G| is invertible in R, then indμρ Sρ

H,V∼= coindμ

τ SτH,V is simple for any simple pair (H,V ).

Remark 6.10. In the case that |G| is invertible in R, we get two different descriptions of SH,V (K)

for K � G. By Corollary 6.9 and proof of Lemma 6.7, we have

SH,V (K) =( ⊕

L�K: L= gH

gV)K

with the maps tNK and rNK given explicitly in Section 2. Also, by Corollary 6.9, we have

SH,V (K) =( ∏

L�K: L= gH

gV

)K

with the maps tNK and rNK given explicitly in Section 2.

7. Semisimplicity

Throughout this section, suppose R is a field in which |G| is a unit. It is well known that theMackey algebra over R is semisimple (see [7,11,12]). The first proof by Thévenaz and Webb [11]is constructive and uses the semisimplicity of the twin functor. In this section we reprove thisresult by giving a shorter proof of the fact that, in this case, the twin functor of a Mackey functoris isomorphic to itself.


Definition 7.1. (See Thévenaz [10].) Let M be a Mackey functor. A subgroup H � G is called aprimordial subgroup for M if defτγ resμ

τ M(H) �= 0.

Recall, without proof, the following lemma.

Lemma 7.2. (See [11, Lemma 9.4].) Let M be a Mackey functor and χ be a subconjugacy closedfamily of subgroups of G. Then,

M = Ker rχ ⊕ Im tχ

where

Ker rχ (K) =⋂

L�K,L∈χ

Ker rKL and Im tχ (K) =

∑L�K,L∈χ

Im tKL

are Mackey subfunctors.

As a consequence of this lemma, we obtain the following decomposition.

Lemma 7.3. Let P = {H0,H1, . . . ,Hn} be the set of all primordial subgroups of a Mackey func-tor M taken up to conjugacy and indexed such that for i < j , no G-conjugate of Hj is containedin Hi . Let Ti denotes the subfunctor of M generated by defτγ resμ

τ M(Hi). Then

M ∼=⊕Hi∈P

Ti

as Mackey functors.

Proof. By Lemma 7.2, we have

M = T0 ⊕ Ker r[H0]

where

Ker r[H0] = Ker rχ and T0 = Im tχ .

Here [H0] is the set of all G-conjugates of H0 and χ is the subconjugacy closure of [H0]. Indeed,we have the equalities since H0 is a minimal subgroup for M . We denote Ker r[H0] by N0. Then,clearly, N0(H0) = 0 and N0(H1) = (defτγ resμ

τ M)(H1). Therefore, by Lemma 7.2 we obtain

N0 = T1 ⊕ N1

where N1 = Ker r[H1]. Note that H1 is a minimal subgroup for N0. Applying the same procedure,we obtain

M =⊕Hi∈P

Ti

as required. �


Let Mi denote the conjugation functor generated by Mi(Hi) = defτγ resμτ M(Hi).

Lemma 7.4. There is an isomorphism of Mackey functors

Ti∼= coindμ

τ infτγ Mi.

Proof. Decomposing Mi into simple summands and applying Corollary 6.9 to each summand,we obtain

indμρ infργ Mi

∼= coindμτ infτγ Mi

where the isomorphism is given by the mark homomorphism. Note that we can decompose Mi

into simple summands since it is clear that the conjugation algebra for G over R is semisimplewhen |G| is invertible in R.

Now consider the following triangle.

Ti

φ

indμρ infργ Mi

β

ψ

coindμτ infτγ Mi

where β is the mark homomorphism and ψ is the induction morphism defined by ψ(tKL ⊗ v) =tKL v for H =G L � K � G and v ∈ Mi(L). The map ψ is a morphism of Mackey functors sinceMi is a minimal subgroup both for Ti and for indμ

ρ infργ Mi (thus ψ commutes with restriction).The map φ is given by

φK(w) = (rKL w

)L�K,L=GH

where K � G and w ∈ Ti(K). Note that ψ is surjective since Ti is generated by its value on theconjugacy class of Hi . Also, since tKL acts as the zero map on infτγ Mi for L �= K , the compositionφ ◦ ψ is the mark homomorphism, that is, the triangle commutes. Moreover since β is injective,the map ψ is also injective. Hence it is an isomorphism. Now it follows that φ = βψ−1 is alsoan isomorphism, as required. �

Finally we are ready to prove the semisimplicity theorem.

Theorem 7.5. (See [11, Theorem 9.1].) The Mackey algebra μR(G) is semisimple if R is a fieldof characteristic coprime to |G|.Proof. Assume the notation of the section. By Lemma 7.3 and Lemma 7.4, we have

M ∼=⊕Hi∈P

coindμτ infτγ Mi.

Inflation and coinduction functors are additive. So decomposing Mi(Hi) into simple RNG(Hi)-modules, we obtain a decomposition of the Mackey functor Ti . But, by Corollary 6.9, the Mackeyfunctor coindμ

τ infτγ Mi is simple if Mi(Hi) is a simple RNG(Hi)-module. �


Corollary 7.6. Let M and N be Mackey functors such that

defτγ resμτ M ∼= defτγ resμ

τ N

as conjugation functors. Then M ∼= N as Mackey functors. In particular,

M ∼= coindμτ infτγ defτγ resμ

τ M.

Proof. This follows from Theorem 7.5 since the simple summands of a Mackey functor M aredetermined by the γ -module defτγ resμ

τ M . Note that the second statement is Corollary 4.4 in [10]and it holds for Mackey functors by Theorem 12.3 of that paper. �Acknowledgment

I wish to thank my supervisor Laurence J. Barker and Ergün Yalçın for their guidance. I amalso grateful to the referee for some important corrections and many detailed suggestions.

References

[1] D.J. Benson, Representations and Cohomology I, Cambridge University Press, Cambridge, UK, 1991.[2] K.S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.[3] S. Bouc, Green Functors and G-Sets, Springer-Verlag, Berlin, 1997.[4] S. Bouc, Résolutions de foncteurs de Mackey, in: Proc. Sympos. Pure Math., vol. 63, 1998, pp. 31–84.[5] S. Bouc, Foncteurs d’ensembles munis d’une double action, J. Algebra 183 (1996) 664–736.[6] R. Boltje, Canonical and explicit Brauer induction in the character ring of a finite group and a generalization for

Mackey functors, Thesis, Universität Augsburg, 1989.[7] R. Boltje, Mackey functors and related structures in representation theory and number theory, Habilitation-Thesis,

Universität Augsburg, 1995.[8] R. Boltje, A general theory of canonical induction formulae, J. Algebra 206 (1998) 293–343.[9] P. Symonds, A splitting principle for modular group representations, Bull. London Math. Soc. 34 (2002) 551–560.

[10] J. Thévenaz, Some remarks on G-functors and the Brauer morphism, J. Reine Angew. Math. (Crelle) 384 (1988)24–56.

[11] J. Thévenaz, P. Webb, Simple Mackey functors, in: Proc. of 2nd International Group Theory Conference, Bres-sonone, 1989, Rend. Circ. Mat. Palermo (2) Suppl. 23 (1990) 299–319.

[12] J. Thévenaz, P. Webb, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995) 1865–1961.

mackey functors, induction from restriction functors and coinduction from transfer functors

Documents